Modeling and Correspondence of Topologically Complex 3D Shapes∗ Ibraheem Alhashim Abstract 3D shape creation and modeling remains a challenging task especially for novice users. Many methods in the field of computer graphics have been pro- posed to automate the often repetitive and precise operations needed during the modeling of detailed shapes. This report surveys different approaches of shape modeling and correspondence especially for shapes exhibiting topological com- plexity. We focus on methods designed to help generate or process shapes with large number of interconnected components often found in man-made shapes. We first discuss a variety of modeling techniques, that leverage existing shapes, in easy to use creative modeling systems. We then discuss possible correspon- dence strategies for topologically different shapes as it is a requirement for such systems. Finally, we look at different shape representations and tools that facili- tate the modification of shape topology and we focus on those particularly useful in free-form 3D modeling. 1 Introduction The creation of quality 3D shapes have been traditionally exclusive to expert design- ers and artists in different fields ranging from industrial design to visual effects. Re- cent advancement in the usability of modeling software, in conjunction with powerful computing hardware, have opened up computer aided design and rapid prototyping for both professional and novice users. Experts often use primitive shape tools, such as line or pen tools, to define 2D outlines or use solid primitives and polygon modi- fication tools for free-form 3D modeling. The more advanced tools that can simplify the modeling process include customized scripts that execute replication or deforma- tion based on some predefined rule set. For a novice user who wants to prototype ideas in her mind, it is much easier to modify on existing shapes than to start from scratch. Furthermore, content creators often draw inspiration from many different arXiv:1506.06855v1 [cs.GR] 23 Jun 2015 partial components of existing examples. The mixing and blending of available parts help in rapid generation of creative designs. Recently available software facilities creative design by utilizing existing large datasets of tailored content or hand made templates. Such systems have been well established in the fields of audio production and in publication and web design. However, in the ∗Technical Report, School of Computing Science, Simon Fraser University, SFU-CMPT TR 2015-55-2 Figure 1: Topology and shape complexity. Complexity of shapes is often related to essential design requirements. For example, the genus zero shape (left) is straightfor- ward to process with most existing graphics algorithms, however, it does not faithfully represents the actual topologically complex shape (right). visual-content field it has yet to expand beyond proposed research projects or spe- cialized 3D character creation software [Poser 2014; DAZ-Studio 2014]. With the enormous availability of visual media as afforded by the Internet, users are now able to draw inspiration form very large datasets of visual content. Recent proposed image composition systems have simplified the conversion of freehand sketches and annota- tions into photomontages extracted from online images [Szeliski 2010]. Similar ideas have also been applied in manual sketch-based 3D part assembly systems that relay on large datasets [Xie et al. 2013]. Such data-driven tools that enable creative design have yet to be widely adapted or commercialized. In order to utilize existing visual content, it is of great importance to automate the analysis and processing of both existing and captured data. A major challenge still facing the fields of computer vision and graphics is the analysis of complex shapes, more often are those of man-made origin. When the topology of a shape exceeds that of a simple curve or a sphere, many shape analysis or processing algorithms are no longer applicable. For example, contour-based methods often only utilize the outer most boundary and omit the analysis of inner holes or disconnected compo- nents. Complications stemming from non-trivial topology of a 3D shape are typically resolved by simplification of the shape to that of sphere or disk topology. However, the complexity in shape topology is often an intrinsic property for a class of shapes (see Figure 1). In this report we investigate existing works on the topic of modeling and analysis of topologically complex 3D shapes. In the first chapter we survey methods for visual content creation by utilizing existing examples via entire or partial blending. Our fo- cus will mainly be on man-made 3D shapes, especially those having widely different structure and topology. In the second chapter we discuss the correspondence prob- lem, more specifically the mapping of very different shapes which is an essential step for blending methods. In the final chapter we look at different shape representations used in shape creation of complex 3D models. We will survey different tools that allow change in shape topology including implicit surfaces, sculpting tools, and other topology-varying methods. 2 Shape Modeling by Blending of Examples For novice users who want to prototype new shape ideas, it is much easier to mod- ify on existing shapes than to start from scratch. Users can start the creative process by retrieving existing visual content with the hope of drawing inspiration. Mixing different pieces or styles often yields interesting and sometimes creative concepts, however, simply swapping components of a shape is quite limiting. A better utiliza- tion that yields many variations is in continuously blending between different shapes akin to the mixing of primary colors to create a versatile color palette. The majority of shape blending or interpolation have been focused on applications in animation where the smooth transition between frames is a desirable property. In the context of shape creation, there are relatively few successful methods that consider the contin- uous blending of the whole shape due to the difficulty of mapping and interpolating topologically different shapes. The more common approach to blending for shape creation has been the mixing of whole parts from different shapes with minor modifi- cations. In this chapter we survey methods related to shape creation from existing visual el- ements. Our focus will be on methods that would work on blending of man-made objects. Later in this chapter we survey methods that mainly work on 3D shapes that exhibit large differences in their overall shape and can have very different topologies. 2.1 Shape Averaging One of the earlier approaches to shape creation in the context of industrial design is the shape averaging method which considers whole shapes [Chen and Parent 1989]. The three stated goals for this work are to help designers predict trends in shape sets, create novel shapes, and stimulate new ideas. While the results presented are rather basic, their pipeline encompasses the major components still in use in more recent shape blending methods; namely, an appropriate shape representation and a corre- spondence scheme. In this work planar polygons are used as the shape representation to blend both 2D and 3D shapes, the later being approximated by slicing the shape into several contours. Finding a meaningful correspondence between the input shapes is a major part of the process (see Figure 2). In their method they define a polygon vertex correspondence that maps shapes, based on minimum distance, and refine the input as necessary. Later efforts [Hui and Li 1998] recognized the need for a better correspondence approach and proposed a feature-based method to match regions that are more reasonably “blendable”. While conceptually the averaging (or blending) of entire shapes can be a powerful tool, current tools available in design software focus mostly on the blending of parts of shapes individually. Tools such as the ”Blend tool” in Adobe Illustrator, Inkscape’s Figure 2: Shape averaging. A set of interpolated and extrapolated results of a Coke bottle from [Chen and Parent 1989] (left). An example of blending two bottles where correspondence is decided based on shape features [Hui and Li 1998] (right). ”Interpolate” tool, or the ”TweenSurfaces” command in Rhino provide users with the ability to generate in-betweens of two given shape segments. However, all such tools perform low-level shape editing operations that still require a lot of practice and manual effort which prohibits their use by novice users. Furthermore, strong assumptions are implicitly made about the compatibility of blended shapes where an exact mapping is needed which in turn limits the ability to blend shapes of different topology. Finally, we note that the averaging of entire shapes at the same time does not necessarily produce as much of interesting variety in the generated forms as that of blending of individual shape parts. This is the case since the number of combinations for the later approach is naturally larger. 2.2 Modeling by Example The work of Funkhouser et al. [Funkhouser et al. 2004] represents one of the first comprehensive systems that allowed for easy shape creation by combining parts from an existing dataset (see Figure 3). The main goal in this work is to develop a 3D modeling tool that requires very little training and user effort making creative shape creation more attainable by novice users. The system simplifies several modeling tasks including part extraction, retrieval, placement and alignment, and stitching of disconnected components. The interactions performed during a modeling session include selecting a starting shape by visual inspection or keyword search, then searching the database for alterna- tives to its parts, and finally applying local edits that identify and stitch the new parts to the active model. The system simplifies part cutting by finding an optimal cut, where cuts along natural seams of the shape are preferred, from a set of user drawn strokes on the shape. The system uses an efficient shape retrieval method, based on an approximate surface distance measure, to query the closest shapes to a query shape from the dataset.